# Load library
library(dplyr)
library(survival)
library(janitor)
library(magrittr)
library(car)
library(ggplot2)
library(tidyverse)
library(broom)
library(MASS)
library(boot)
#print(citation("survival"), bibtex=TRUE)Cox regression modeling of survival after chemotherapy for colon cancer
Data: Survival after chemotherapy for Stage B/C colon cancer [1] [2]
Description
These are data from one of the first successful trials of adjuvant chemotherapy for colon cancer. Levamisole is a low-toxicity compound previously used to treat worm infestations in animals; 5-FU is a moderately toxic (as these things go) chemotherapy agent. There are two records per person, one for recurrence and one for death.
The purpose of this project is to compare survival between the untreated (Obs) group vs those treated with amisole (Lev), or amisole + 5-FU.
Column names:
| id: | id |
| study: | 1 for all patients |
| rx: | Treatment - Obs(ervation), Lev(amisole), Lev(amisole)+5-FU |
| sex: | 1=male |
| age: | in years |
| obstruct: | obstruction of colon by tumour |
| perfor: | perforation of colon |
| adhere: | adherence to nearby organs |
| nodes: | number of lymph nodes with detectable cancer |
| time: | days until event or censoring |
| status: | censoring status |
| differ: | differentiation of tumour (1=well, 2=moderate, 3=poor) |
| extent: | Extent of local spread (1=submucosa, 2=muscle, 3=serosa, 4=contiguous structures) |
| surg: | time from surgery to registration (0=short, 1=long) |
| node4: | more than 4 positive lymph nodes |
| etype: | event type: 1=recurrence,2=death |
#Load data
colon <- as_tibble(colon)
head(colon)# A tibble: 6 × 16
id study rx sex age obstruct perfor adhere nodes status differ
<dbl> <dbl> <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1 Lev+5FU 1 43 0 0 0 5 1 2
2 1 1 Lev+5FU 1 43 0 0 0 5 1 2
3 2 1 Lev+5FU 1 63 0 0 0 1 0 2
4 2 1 Lev+5FU 1 63 0 0 0 1 0 2
5 3 1 Obs 0 71 0 0 1 7 1 2
6 3 1 Obs 0 71 0 0 1 7 1 2
# ℹ 5 more variables: extent <dbl>, surg <dbl>, node4 <dbl>, time <dbl>,
# etype <dbl>Since the current analysis is focused on survival, filter data to death as the event type. This will create a data table with one row per individual.
colon_surv <- colon%>%filter(etype == 1) I. Exploratory data analysis
Check missing values
na_counts <- sapply(colon_surv, function(x)sum(is.na(x)))
na_counts id study rx sex age obstruct perfor adhere
0 0 0 0 0 0 0 0
nodes status differ extent surg node4 time etype
18 0 23 0 0 0 0 0 # replace NAs with mode
table(colon_surv$differ)
1 2 3
93 663 150 mode(colon_surv$differ)[1] "numeric"median(colon_surv$nodes, na.rm= TRUE)[1] 2colon_surv$differ <- if_else(is.na(colon_surv$differ), 2,colon_surv$differ)
colon_surv$nodes <- if_else(is.na(colon_surv$nodes), 2,colon_surv$nodes)Insight: only nodes and differ columns have NA values. Replacing the 23 NAs in differ column with mode, and replace NAs in nodes with median.
Evaluate continuous variables
# age
hist(colon_surv$age)
hist(colon_surv$nodes)
hist(colon_surv$time)
Insight: Age is normally distributed. Number of nodes is skewed to the right. Time is fairly normally distributed with most the individuals having event time between 500-3000 days.
Evaluate nodes column to investigate outliers
t <- colon_surv%>%filter(node4 ==1) # samples with more than 4 positive lymph nodes
hist(t$nodes) 
Insight: samples with greater than 4 lymph nodes have less than 5 count in nodes column, so the two columns are not consistent. Therefore, nodes column will not be used for further analysis.
Evaluate categorical variables
summary_table <- colon_surv%>%summarise(count =n(),
male = sum(sex),
median_age = median(age),
ct_perforation = sum(perfor),
ct_adherence_nerby_organ = sum(adhere), censored = sum(status))
summary_table# A tibble: 1 × 6
count male median_age ct_perforation ct_adherence_nerby_organ censored
<int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 929 484 61 27 135 468Insight: Total number of participants: 929. About half of the participants are male and about half were censored, while the other half died.
# rename categorical columns to make them descriptive
colon_surv <- colon_surv%>%mutate(differentiation = case_when(differ == 1 ~ "well",
differ == 2 ~ "moderate",
differ == 3 ~ "poor"),
local_spread = case_when(extent == 1 ~ "submucosa",
extent == 2 ~ "muscle",
extent == 3 ~ "serosa",
extent == 4 ~ "contiguous"),
surg_to_reg_time = case_when(surg == 0~ "short",
surg == 1 ~ "long"))Frequency tables for categorical variables
# frequency tables for categorical variables
# Tumor differentiation
colon_surv %>%
tabyl(differentiation, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() differentiation Obs Lev Lev+5FU
moderate 74.9% (236) 73.9% (229) 72.7% (221)
poor 16.5% (52) 14.2% (44) 17.8% (54)
well 8.6% (27) 11.9% (37) 9.5% (29)# extent of local spread
colon_surv %>%
tabyl(local_spread, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() local_spread Obs Lev Lev+5FU
contiguous 6.3% (20) 3.9% (12) 3.6% (11)
muscle 12.1% (38) 11.6% (36) 10.5% (32)
serosa 79.0% (249) 83.5% (259) 82.6% (251)
submucosa 2.5% (8) 1.0% (3) 3.3% (10)# colum obstruction
colon_surv %>%
tabyl(obstruct, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() obstruct Obs Lev Lev+5FU
0 80.0% (252) 79.7% (247) 82.2% (250)
1 20.0% (63) 20.3% (63) 17.8% (54)# colon perforation
colon_surv %>%
tabyl(perfor, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() perfor Obs Lev Lev+5FU
0 97.1% (306) 96.8% (300) 97.4% (296)
1 2.9% (9) 3.2% (10) 2.6% (8)# Adherance to nearby organs
colon_surv %>%
tabyl(adhere, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() adhere Obs Lev Lev+5FU
0 85.1% (268) 84.2% (261) 87.2% (265)
1 14.9% (47) 15.8% (49) 12.8% (39)# extent of local tumor spread
colon_surv %>%
tabyl(local_spread, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() local_spread Obs Lev Lev+5FU
contiguous 6.3% (20) 3.9% (12) 3.6% (11)
muscle 12.1% (38) 11.6% (36) 10.5% (32)
serosa 79.0% (249) 83.5% (259) 82.6% (251)
submucosa 2.5% (8) 1.0% (3) 3.3% (10)# More than 4 lymph nodes with cancer
colon_surv %>%
tabyl(node4, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() node4 Obs Lev Lev+5FU
0 72.4% (228) 71.3% (221) 74.0% (225)
1 27.6% (87) 28.7% (89) 26.0% (79)# time from surgery to registration
colon_surv %>%
tabyl(surg, rx) %>%
adorn_percentages("col") %>%
adorn_pct_formatting(digits = 1) %>%
adorn_ns() surg Obs Lev Lev+5FU
0 71.1% (224) 74.2% (230) 75.0% (228)
1 28.9% (91) 25.8% (80) 25.0% (76)Summary statistics grouped by treatment
summary_table <- colon_surv%>%group_by(rx)%>%summarise(count =n(),
male = sum(sex),
median_age = median(age),
ct_perforation = sum(perfor),
ct_adherence_nerby_organ = sum(adhere),
perc_male = (male/count)*100,
iqr_age = IQR(age))
summary_table# A tibble: 3 × 8
rx count male median_age ct_perforation ct_adherence_nerby_o…¹ perc_male
<fct> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 Obs 315 166 60 9 47 52.7
2 Lev 310 177 61 10 49 57.1
3 Lev+5FU 304 141 62 8 39 46.4
# ℹ abbreviated name: ¹ct_adherence_nerby_organ
# ℹ 1 more variable: iqr_age <dbl>Insight: Each treatment group had about 300 participants. Median age, number of participants with perforation and adherence are similar between the three groups.
II. Table 1: Description of the study population
| Observation (%) | Amisole (%) | Amisole + 5-FU (%) | ||
|---|---|---|---|---|
| N=315 | N=310 | N=304 | ||
| Demographics | ||||
| Male | 166 (52.3) | 177 (57.1) | 141 | |
| Median age (years) [IQR] | 60 [53,68] | 61 [53,69] | 61 [52,70] | |
| Cancer characteristics | ||||
| Colon obstruction | 63 (20.0) | 63 (20.3) | 54 (17.8) | |
| Colon perforation | 9 (2.9) | 10 (3.2) | 8 (2.6) | |
| Adherence to nearby organs | 47 (14.9) | 49 (15.8) | 39 (12.8) | |
| Differentiation of tumor | ||||
| Well | 27 (8.6) | 37 (11.9) | 29 (9.5) | |
| Moderate | 236 (74.9) | 229 (73.9) | 221 (72.7) | |
| Poor | 52 (16.5) | 44 (14.2) | 54 (17.8) | |
| Extent of local spread | ||||
| Contiguous | 20 (6.3) | 12 (3.9) | 11 (3.6) | |
| Muscle | 38 (12.1) | 36 (11.6) | 32 (10.5) | |
| Serosa | 249 (79.0) | 259 (83.5) | 251 (82.6) | |
| Submucosa | 8 (2.5) | 3 (1.0) | 10 (3.3) | |
| More than 4 lymph nodes with cancer | Yes | 87 (27.6) | 89 (28.7) | 79 (26.0) |
| Short time from surgery to registration (%) | Yes | 91 (28.9) | 80 (25.8) | 76 (25.0) |
III. Methods
The Cox proportional hazards model was used to model the relationship between survival time and different lung cancer treatments. In particlular the survival time will be compared between the untreated group (observation) vs. those treated with amisole (Lev), or amisole + 5-FU. The Cox regression model was chosen for this study because it is useful for studying association between survival time of patients and predictors and allows estimating the relative risk or hazard ratios due to the covariates, i.e., treatment status. The time (in days) until event, i.e, death, will be modeled as a function of treatment and other variables, including age, sex, and various tumor characteristics. Significant predictors were included in the final model.
Statistical analysis
The R statistical software version 4.3.2 [3] was used for all analysis. The Survival package was used to construct the Cox regression model [2] [1].
Cox regression model is based on the hazard function \(h_x(t)\) with covariates at time t given by:
\(h_x(t)=h_0(t)\exp(\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_p)\)
Where:
\(h_x(t)\) is the hazard function
\(h_0(t)\) is the baseline hazard function
\(\beta_1x_1 + \beta_2x_2 + \dots +\beta_p x_p\) represent the linear combination of covariates and their coefficient
The hazards for the observation vs. amisole (Lev), or amisole + 5-FU group with covariate values x1 and x2 are given by: \(hx_1(t)=h_0(t)\exp(\beta_1x_1)\) and \(hx_2(t)=h_0(t)\exp(\beta_2x_2)\), respectively
The hazard ratio is expressed as: HR = \(hx_2(t)\) / \(hx_1(t)\) = \(\exp[\beta(x_2-x_1)]\)
The Schoenfeld residual plot was constructed to test Cox proportional hazards assumption. When the proportional hazards assumpiton was not met for any of the covariates, stratification approach was explored. The Survminer [4] package was used to plot the Kaplan-Meier curve to visualize the survival probability over time for each treatment group.
Multicolinearity was tested using Variant Inflation Factor (VIF) calculated using MASS package [5].
The R MASS package was used for Stepwise model selection, using “both” forward and backward variable selection [5]. For Stepwise selection, stepAIC() function uses AIC (Akaike Information Criterion) as the measure to add or remove predictors from the model. Model performance was evaluated using 100-fold cross-validation using the boot package [6] [7].
IV. Analysis: Cox regression model
Survival curve
# Create new incremental count id
colon_surv$idcount <- c(1:length(colon_surv$id))
# Order by survival time and create an order variable:
colon_surv <- colon_surv[order(-colon_surv$time, colon_surv$status),]
colon_surv$order <- c(1:length(colon_surv$idcount))
ggplot(data=colon_surv, aes(x=time, y=order)) +
geom_rect(xmin=23,xmax=colon_surv$time,ymin=colon_surv$order,ymax=colon_surv$order+1, colour="lightgray") +
geom_rect(xmin=colon_surv$time-2,xmax=colon_surv$time,ymin=colon_surv$order,ymax=colon_surv$order+1,
fill=factor(colon_surv$status+1)) +
geom_vline(xintercept= 1976,linetype="solid") +
scale_x_continuous(breaks=seq(20,3330,650)) +
geom_text(aes(2600, 750, label="Median Survival Time")) +
xlab("Survival Time (Days)") + ylab("Participants (ordered by survival time)") +
ggtitle("Survival Times for Participant") +
theme_classic() +
theme(legend.position="none",
panel.grid.major=element_blank(),
panel.grid.minor=element_blank(),
panel.background=element_blank(),
axis.line.x = element_line(color = "black"),
axis.line.y = element_line(color = "black"))
Survival curve stratified by treatment group
library(survminer)
library(survival)
# Estimate the median survival time among the three groups
survfit(Surv(time,status) ~ rx, data = colon_surv)Call: survfit(formula = Surv(time, status) ~ rx, data = colon_surv)
n events median 0.95LCL 0.95UCL
rx=Obs 315 177 1236 803 2036
rx=Lev 310 172 1183 797 2067
rx=Lev+5FU 304 119 NA NA NA# count the number of events after 2080 days, which is the median survival time among the observation group
tt <- colon_surv%>%filter(time > 2083)%>% group_by(rx)%>%summarise(ct = n(),
death = sum(status))
# Plot survival curve
fit <- survfit(Surv(time,status) ~ rx, data = colon_surv)
ggsurvplot(fit, data=colon_surv, risk.table = TRUE)
# Estimate the probability of surviving beyond 3000 days
summary(survfit(Surv(time, status) ~ rx, data = colon_surv), times = 3000)Call: survfit(formula = Surv(time, status) ~ rx, data = colon_surv)
rx=Obs
time n.risk n.event survival std.err lower 95% CI
3.00e+03 5.00e+00 1.77e+02 4.07e-01 3.35e-02 3.47e-01
upper 95% CI
4.79e-01
rx=Lev
time n.risk n.event survival std.err lower 95% CI
3.00e+03 4.00e+00 1.72e+02 4.33e-01 2.87e-02 3.80e-01
upper 95% CI
4.93e-01
rx=Lev+5FU
time n.risk n.event survival std.err lower 95% CI
3.00e+03 7.00e+00 1.19e+02 5.99e-01 2.86e-02 5.46e-01
upper 95% CI
6.58e-01 # compare significant differences in survival times between the three groups
survdiff(Surv(time, status)~ rx, data = colon_surv)Call:
survdiff(formula = Surv(time, status) ~ rx, data = colon_surv)
N Observed Expected (O-E)^2/E (O-E)^2/V
rx=Obs 315 177 151 4.62 6.82
rx=Lev 310 172 149 3.69 5.41
rx=Lev+5FU 304 119 169 14.69 23.04
Chisq= 23.1 on 2 degrees of freedom, p= 1e-05 Insight: Based on the survival curve, the mediant survival time for the observation group is 2083 days. However, the median survival of Lev and Lev+5Fu group cannot be estimated, because there are too few events after 2083 days, which is the median survival time in the observation group.
The time for 50% survival probability of the group treated with Lev+5Fu is over 3000 days while the survival time for the observation and Lev group is around 2080 days. The probability of surviving to 3000 days among the Lev+5FU group is 56% (95% CI: 50-63), compared to 41% among the observation group.
The survival time is significantly different (P=0.003) between the three groups.
Insight: None of the variables have VIF values above 5, therefore there is no multicollinearity
Cox regression models
# Subset data for modeling
df <- colon_surv%>%dplyr::select(!c(id,study,etype,differ, extent,surg_to_reg_time, idcount, order, nodes))Base Model
m0 <- coxph(Surv(time, status) ~ 1, data = df)
summary_m0 = summary(m0)
c_index_m0 <- concordance(m0)
cat("Concordance of the base model:",c_index_m0$concordance)Concordance of the base model: 0.5Univariate analysis
#| echo: true
#| message: false
#| warning: false
# Univariate analysis
m1 <- coxph(Surv(time, status) ~ rx, data = df)
summary(m1)Call:
coxph(formula = Surv(time, status) ~ rx, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev -0.01512 0.98499 0.10708 -0.141 0.888
rxLev+5FU -0.51209 0.59924 0.11863 -4.317 1.58e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 0.9850 1.015 0.7985 1.2150
rxLev+5FU 0.5992 1.669 0.4749 0.7561
Concordance= 0.554 (se = 0.013 )
Likelihood ratio test= 24.34 on 2 df, p=5e-06
Wald test = 22.58 on 2 df, p=1e-05
Score (logrank) test = 23.07 on 2 df, p=1e-05c_index_m1 <- concordance(m1)
cat("Concordance of the univariate model:",c_index_m1$concordance)Concordance of the univariate model: 0.5544598anova(m0, m1) # Addition of rx variable significantly improved base modelAnalysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ 1
Model 2: ~ rx
loglik Chisq Df Pr(>|Chi|)
1 -3040.3
2 -3028.1 24.343 2 5.175e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Insight: the coefficient of Lev is not significant, suggesting that there is no evidence that this treatment affects survival time compared to observation. however Lev+5Fu is significant (p=0.00175), indicating that the treatment Lev +5Fu has a statistically significant effect on survival time compared to the reference group. The negative sign indicates that this treatment group has a lower hazard and likely a longer survival time.
The hazard ratio for Lex+5FU (0.690), indicating the risk of death is about 31% lower compared to the observation group.
The p-values indicate that the model is significant.
Test the Cox proportional hazard assumption of m1
cox.zph(m1) chisq df p
rx 0.301 2 0.86
GLOBAL 0.301 2 0.86zph_test <- cox.zph(m1)
print(zph_test) chisq df p
rx 0.301 2 0.86
GLOBAL 0.301 2 0.86# plot the Schoenfeld residuals
plot(zph_test)
Insight: The Schoenfeld residal plot shows that the residuals are scattered randomly and the smooth trend line is horizontal near 0. This suggests that the hazard ratio for rx (treatment status) is constant over time and the proportional hazard assumption is met. The global p-value is >0.05, indicating that the the assumption is met.
Multivariate analysis
# Include all variables to determine which predictors are significant.
names(df) [1] "rx" "sex" "age" "obstruct"
[5] "perfor" "adhere" "status" "surg"
[9] "node4" "time" "differentiation" "local_spread" # multivariate analysis
m2 <- coxph(Surv(time, status) ~ rx+ age + sex + perfor + adhere + surg + obstruct + differentiation + node4+
local_spread, data = df)
summary(m2)Call:
coxph(formula = Surv(time, status) ~ rx + age + sex + perfor +
adhere + surg + obstruct + differentiation + node4 + local_spread,
data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev -0.000276 0.999724 0.108407 -0.003 0.99797
rxLev+5FU -0.513501 0.598397 0.119205 -4.308 1.65e-05 ***
age -0.002809 0.997195 0.003923 -0.716 0.47400
sex -0.080319 0.922822 0.093672 -0.857 0.39120
perfor 0.188904 1.207925 0.255841 0.738 0.46029
adhere 0.168813 1.183898 0.128257 1.316 0.18811
surg 0.247327 1.280597 0.101407 2.439 0.01473 *
obstruct 0.199681 1.221013 0.115875 1.723 0.08484 .
differentiationpoor 0.338549 1.402910 0.121115 2.795 0.00519 **
differentiationwell 0.024189 1.024484 0.162583 0.149 0.88173
node4 0.842349 2.321814 0.097335 8.654 < 2e-16 ***
local_spreadmuscle -1.048500 0.350463 0.259637 -4.038 5.38e-05 ***
local_spreadserosa -0.478763 0.619549 0.199176 -2.404 0.01623 *
local_spreadsubmucosa -1.113624 0.328367 0.491639 -2.265 0.02351 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 0.9997 1.0003 0.8084 1.2364
rxLev+5FU 0.5984 1.6711 0.4737 0.7559
age 0.9972 1.0028 0.9896 1.0049
sex 0.9228 1.0836 0.7680 1.1088
perfor 1.2079 0.8279 0.7316 1.9944
adhere 1.1839 0.8447 0.9207 1.5223
surg 1.2806 0.7809 1.0498 1.5622
obstruct 1.2210 0.8190 0.9729 1.5323
differentiationpoor 1.4029 0.7128 1.1065 1.7788
differentiationwell 1.0245 0.9761 0.7449 1.4090
node4 2.3218 0.4307 1.9186 2.8098
local_spreadmuscle 0.3505 2.8534 0.2107 0.5830
local_spreadserosa 0.6195 1.6141 0.4193 0.9154
local_spreadsubmucosa 0.3284 3.0454 0.1253 0.8607
Concordance= 0.669 (se = 0.012 )
Likelihood ratio test= 149.8 on 14 df, p=<2e-16
Wald test = 150.7 on 14 df, p=<2e-16
Score (logrank) test = 160.4 on 14 df, p=<2e-16c_index_m2 <- concordance(m2)
cat("Concordance of the multivariate model:",c_index_m2$concordance)Concordance of the multivariate model: 0.6694041# Determine significant predictors
anova(m2)Analysis of Deviance Table
Cox model: response is Surv(time, status)
Terms added sequentially (first to last)
loglik Chisq Df Pr(>|Chi|)
NULL -3040.3
rx -3028.1 24.3435 2 5.175e-06 ***
age -3026.8 2.5853 1 0.1078626
sex -3026.2 1.3125 1 0.2519380
perfor -3025.2 1.8472 1 0.1741088
adhere -3022.9 4.7199 1 0.0298156 *
surg -3020.3 5.1721 1 0.0229519 *
obstruct -3019.1 2.3959 1 0.1216554
differentiation -3012.9 12.3380 2 0.0020933 **
node4 -2975.3 75.1393 1 < 2.2e-16 ***
local_spread -2965.4 19.9397 3 0.0001747 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(m0, m1, m2)Analysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ 1
Model 2: ~ rx
Model 3: ~ rx + age + sex + perfor + adhere + surg + obstruct + differentiation + node4 + local_spread
loglik Chisq Df Pr(>|Chi|)
1 -3040.3
2 -3028.1 24.343 2 5.175e-06 ***
3 -2965.4 125.450 12 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Insight: When all variables are included in the model, the anova test indicates that rx, adhere, surg, obstruct, differentiation, node4 and local spread are significant predictors. Additionally, model concordance did not improve when removing predictors that were not significant in m2
The concordance of the multivariable model, 0.674, is higher than the univariate model (m1, concordance =0.53), suggesting that the multivariate model is a better fit model.
Evaluate multicollinearity using Variance Inflation Factor (VIF)
vif <- vif(m2)
print(vif) GVIF Df GVIF^(1/(2*Df))
rx 1.033078 2 1.008169
age 1.036544 1 1.018108
sex 1.025735 1 1.012786
perfor 1.071324 1 1.035048
adhere 1.111746 1 1.054393
surg 1.012413 1 1.006187
obstruct 1.047540 1 1.023494
differentiation 1.052692 2 1.012920
node4 1.035800 1 1.017742
local_spread 1.107687 3 1.017192Evaluate significance of predictors. Model survival while including different cancer characteristics as predictors separately to identify significance predictors.
# model including all variables
m2 <- coxph(Surv(time, status) ~ rx+ age + sex + perfor + adhere + surg + obstruct + differentiation + node4+
local_spread, data = df)
# Treatment
m2a <- coxph(Surv(time, status) ~ rx, data = df) # significant
summary(m2a)Call:
coxph(formula = Surv(time, status) ~ rx, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev -0.01512 0.98499 0.10708 -0.141 0.888
rxLev+5FU -0.51209 0.59924 0.11863 -4.317 1.58e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 0.9850 1.015 0.7985 1.2150
rxLev+5FU 0.5992 1.669 0.4749 0.7561
Concordance= 0.554 (se = 0.013 )
Likelihood ratio test= 24.34 on 2 df, p=5e-06
Wald test = 22.58 on 2 df, p=1e-05
Score (logrank) test = 23.07 on 2 df, p=1e-05# Demographics
m2b <- coxph(Surv(time, status) ~ age + sex, data = df) # not significant
summary(m2b)Call:
coxph(formula = Surv(time, status) ~ age + sex, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
age -0.006715 0.993308 0.003880 -1.731 0.0835 .
sex -0.084298 0.919157 0.092483 -0.911 0.3620
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
age 0.9933 1.007 0.9858 1.001
sex 0.9192 1.088 0.7668 1.102
Concordance= 0.527 (se = 0.014 )
Likelihood ratio test= 3.77 on 2 df, p=0.2
Wald test = 3.82 on 2 df, p=0.1
Score (logrank) test = 3.82 on 2 df, p=0.1# cancer characteristics
m2c <- coxph(Surv(time, status) ~ perfor + adhere + obstruct, data = df) # adhere and obstruct are significant
summary(m2c)Call:
coxph(formula = Surv(time, status) ~ perfor + adhere + obstruct,
data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
perfor 0.1729 1.1888 0.2554 0.677 0.4983
adhere 0.2923 1.3395 0.1242 2.353 0.0186 *
obstruct 0.2194 1.2454 0.1146 1.914 0.0556 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
perfor 1.189 0.8412 0.7207 1.961
adhere 1.340 0.7465 1.0500 1.709
obstruct 1.245 0.8030 0.9948 1.559
Concordance= 0.539 (se = 0.011 )
Likelihood ratio test= 10.61 on 3 df, p=0.01
Wald test = 11.4 on 3 df, p=0.01
Score (logrank) test = 11.49 on 3 df, p=0.009# Differentiation of tumor
m2d <- coxph(Surv(time, status) ~ differentiation, data = df) # significant
summary(m2d)Call:
coxph(formula = Surv(time, status) ~ differentiation, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
differentiationpoor 0.4460 1.5621 0.1199 3.721 0.000198 ***
differentiationwell -0.0523 0.9490 0.1603 -0.326 0.744262
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
differentiationpoor 1.562 0.6402 1.2350 1.976
differentiationwell 0.949 1.0537 0.6931 1.299
Concordance= 0.541 (se = 0.011 )
Likelihood ratio test= 13.35 on 2 df, p=0.001
Wald test = 14.68 on 2 df, p=6e-04
Score (logrank) test = 14.93 on 2 df, p=6e-04# Extent of local spread
m2e <- coxph(Surv(time, status) ~ local_spread, data = df) # significant
summary(m2e)Call:
coxph(formula = Surv(time, status) ~ local_spread, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
local_spreadmuscle -1.2391 0.2896 0.2533 -4.891 1e-06 ***
local_spreadserosa -0.5663 0.5676 0.1927 -2.939 0.00329 **
local_spreadsubmucosa -1.5668 0.2087 0.4846 -3.233 0.00122 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
local_spreadmuscle 0.2896 3.453 0.17628 0.4759
local_spreadserosa 0.5676 1.762 0.38909 0.8281
local_spreadsubmucosa 0.2087 4.791 0.08074 0.5395
Concordance= 0.551 (se = 0.009 )
Likelihood ratio test= 32.65 on 3 df, p=4e-07
Wald test = 29.3 on 3 df, p=2e-06
Score (logrank) test = 30.96 on 3 df, p=9e-07# short time from surgery to registration
m2f <- coxph(Surv(time, status) ~ surg, data = df) # significant
summary(m2f)Call:
coxph(formula = Surv(time, status) ~ surg, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
surg 0.2549 1.2903 0.1008 2.529 0.0114 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
surg 1.29 0.775 1.059 1.572
Concordance= 0.523 (se = 0.01 )
Likelihood ratio test= 6.17 on 1 df, p=0.01
Wald test = 6.39 on 1 df, p=0.01
Score (logrank) test = 6.43 on 1 df, p=0.01# include predictors significant in the model which included all predictors (m2)
m3 <- coxph(Surv(time, status) ~ rx + surg + obstruct + differentiation + node4
+ local_spread, data = df)
summary(m3)Call:
coxph(formula = Surv(time, status) ~ rx + surg + obstruct + differentiation +
node4 + local_spread, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev 0.0007498 1.0007501 0.1081783 0.007 0.99447
rxLev+5FU -0.5078980 0.6017591 0.1192067 -4.261 2.04e-05 ***
surg 0.2513711 1.2857872 0.1013684 2.480 0.01315 *
obstruct 0.2215931 1.2480634 0.1142954 1.939 0.05253 .
differentiationpoor 0.3531967 1.4236111 0.1205339 2.930 0.00339 **
differentiationwell 0.0340964 1.0346843 0.1617970 0.211 0.83309
node4 0.8455285 2.3292085 0.0967209 8.742 < 2e-16 ***
local_spreadmuscle -1.1345405 0.3215698 0.2552751 -4.444 8.81e-06 ***
local_spreadserosa -0.5565149 0.5732032 0.1935786 -2.875 0.00404 **
local_spreadsubmucosa -1.1924159 0.3034872 0.4872601 -2.447 0.01440 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 1.0008 0.9993 0.8096 1.2371
rxLev+5FU 0.6018 1.6618 0.4764 0.7601
surg 1.2858 0.7777 1.0541 1.5684
obstruct 1.2481 0.8012 0.9976 1.5614
differentiationpoor 1.4236 0.7024 1.1241 1.8030
differentiationwell 1.0347 0.9665 0.7535 1.4208
node4 2.3292 0.4293 1.9270 2.8154
local_spreadmuscle 0.3216 3.1097 0.1950 0.5304
local_spreadserosa 0.5732 1.7446 0.3922 0.8377
local_spreadsubmucosa 0.3035 3.2950 0.1168 0.7887
Concordance= 0.667 (se = 0.012 )
Likelihood ratio test= 146.1 on 10 df, p=<2e-16
Wald test = 147.8 on 10 df, p=<2e-16
Score (logrank) test = 157 on 10 df, p=<2e-16c_index_m3 <- concordance(m3)
cat("Concordance of the multivariate model2:",c_index_m3$concordance)Concordance of the multivariate model2: 0.6667876anova(m0, m1, m2, m2a, m2b, m2c, m2d, m2e, m2f, m3)Analysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ 1
Model 2: ~ rx
Model 3: ~ rx + age + sex + perfor + adhere + surg + obstruct + differentiation + node4 + local_spread
Model 4: ~ rx
Model 5: ~ age + sex
Model 6: ~ perfor + adhere + obstruct
Model 7: ~ differentiation
Model 8: ~ local_spread
Model 9: ~ surg
Model 10: ~ rx + surg + obstruct + differentiation + node4 + local_spread
loglik Chisq Df Pr(>|Chi|)
1 -3040.3
2 -3028.1 24.3435 2 5.175e-06 ***
3 -2965.4 125.4499 12 < 2.2e-16 ***
4 -3028.1 125.4499 12 < 2.2e-16 ***
5 -3038.4 20.5731 0 < 2.2e-16 ***
6 -3035.0 6.8362 1 0.008933 **
7 -3033.6 2.7482 1 0.097367 .
8 -3023.9 19.2935 1 1.121e-05 ***
9 -3037.2 26.4771 2 1.781e-06 ***
10 -2967.2 139.8791 9 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Insight: rx, adhere, surg, obstruct, differentiation and local_spread are significant predictors. However, the model concordance is low (~0.5) when each was included separately. Model m2, which included all predictors, followed by model 3 (including selected significant predictors) have the highest concordance.
Perform Stepwise variable selection:
library(MASS) # for stepwise regression
#### Use the MASS package stepAIC() function for stepwise selection by using AIC (Akaike Information Criterion) as the measure to add or remove predictors from the model.
# model including all variables
m2 <- coxph(Surv(time, status) ~ rx+ age + sex + perfor + adhere + surg + obstruct + differentiation + node4+
local_spread, data = df)
# stepwise selection
stepwise_model <- stepAIC(m2, direction = "both")Start: AIC=5958.76
Surv(time, status) ~ rx + age + sex + perfor + adhere + surg +
obstruct + differentiation + node4 + local_spread
Df AIC
- age 1 5957.3
- perfor 1 5957.3
- sex 1 5957.5
- adhere 1 5958.4
<none> 5958.8
- obstruct 1 5959.6
- differentiation 2 5962.1
- surg 1 5962.5
- local_spread 3 5972.7
- rx 2 5979.7
- node4 1 6026.0
Step: AIC=5957.27
Surv(time, status) ~ rx + sex + perfor + adhere + surg + obstruct +
differentiation + node4 + local_spread
Df AIC
- perfor 1 5955.9
- sex 1 5956.0
- adhere 1 5956.8
<none> 5957.3
- obstruct 1 5958.4
+ age 1 5958.8
- differentiation 2 5960.5
- surg 1 5961.0
- local_spread 3 5971.5
- rx 2 5978.2
- node4 1 6026.0
Step: AIC=5955.87
Surv(time, status) ~ rx + sex + adhere + surg + obstruct + differentiation +
node4 + local_spread
Df AIC
- sex 1 5954.5
- adhere 1 5955.8
<none> 5955.9
+ perfor 1 5957.3
+ age 1 5957.3
- obstruct 1 5957.4
- differentiation 2 5959.2
- surg 1 5959.6
- local_spread 3 5970.6
- rx 2 5976.7
- node4 1 6024.2
Step: AIC=5954.51
Surv(time, status) ~ rx + adhere + surg + obstruct + differentiation +
node4 + local_spread
Df AIC
- adhere 1 5954.5
<none> 5954.5
+ sex 1 5955.9
+ age 1 5955.9
+ perfor 1 5956.0
- obstruct 1 5956.2
- differentiation 2 5957.8
- surg 1 5958.2
- local_spread 3 5969.3
- rx 2 5975.0
- node4 1 6023.5
Step: AIC=5954.5
Surv(time, status) ~ rx + surg + obstruct + differentiation +
node4 + local_spread
Df AIC
<none> 5954.5
+ adhere 1 5954.5
+ perfor 1 5955.6
+ sex 1 5955.8
+ age 1 5956.1
- obstruct 1 5956.1
- surg 1 5958.4
- differentiation 2 5958.5
- local_spread 3 5971.1
- rx 2 5975.0
- node4 1 6022.9summary(stepwise_model)Call:
coxph(formula = Surv(time, status) ~ rx + surg + obstruct + differentiation +
node4 + local_spread, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev 0.0007498 1.0007501 0.1081783 0.007 0.99447
rxLev+5FU -0.5078980 0.6017591 0.1192067 -4.261 2.04e-05 ***
surg 0.2513711 1.2857872 0.1013684 2.480 0.01315 *
obstruct 0.2215931 1.2480634 0.1142954 1.939 0.05253 .
differentiationpoor 0.3531967 1.4236111 0.1205339 2.930 0.00339 **
differentiationwell 0.0340964 1.0346843 0.1617970 0.211 0.83309
node4 0.8455285 2.3292085 0.0967209 8.742 < 2e-16 ***
local_spreadmuscle -1.1345405 0.3215698 0.2552751 -4.444 8.81e-06 ***
local_spreadserosa -0.5565149 0.5732032 0.1935786 -2.875 0.00404 **
local_spreadsubmucosa -1.1924159 0.3034872 0.4872601 -2.447 0.01440 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 1.0008 0.9993 0.8096 1.2371
rxLev+5FU 0.6018 1.6618 0.4764 0.7601
surg 1.2858 0.7777 1.0541 1.5684
obstruct 1.2481 0.8012 0.9976 1.5614
differentiationpoor 1.4236 0.7024 1.1241 1.8030
differentiationwell 1.0347 0.9665 0.7535 1.4208
node4 2.3292 0.4293 1.9270 2.8154
local_spreadmuscle 0.3216 3.1097 0.1950 0.5304
local_spreadserosa 0.5732 1.7446 0.3922 0.8377
local_spreadsubmucosa 0.3035 3.2950 0.1168 0.7887
Concordance= 0.667 (se = 0.012 )
Likelihood ratio test= 146.1 on 10 df, p=<2e-16
Wald test = 147.8 on 10 df, p=<2e-16
Score (logrank) test = 157 on 10 df, p=<2e-16# Multivariate model including variables selected based on stepwise variable selection. The same variables were significant based on anova test of the model that included all variables.
m4 <- coxph(Surv(time, status) ~ rx + age + surg + obstruct +
differentiation + node4 + local_spread, data = df)
summary(m4)Call:
coxph(formula = Surv(time, status) ~ rx + age + surg + obstruct +
differentiation + node4 + local_spread, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev 0.001883 1.001885 0.108199 0.017 0.98612
rxLev+5FU -0.507444 0.602033 0.119200 -4.257 2.07e-05 ***
age -0.002617 0.997387 0.003923 -0.667 0.50479
surg 0.251935 1.286513 0.101365 2.485 0.01294 *
obstruct 0.213610 1.238140 0.114930 1.859 0.06308 .
differentiationpoor 0.355638 1.427091 0.120577 2.949 0.00318 **
differentiationwell 0.031717 1.032226 0.161856 0.196 0.84464
node4 0.839740 2.315764 0.097116 8.647 < 2e-16 ***
local_spreadmuscle -1.125910 0.324357 0.255575 -4.405 1.06e-05 ***
local_spreadserosa -0.551200 0.576258 0.193725 -2.845 0.00444 **
local_spreadsubmucosa -1.197289 0.302012 0.487303 -2.457 0.01401 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 1.0019 0.9981 0.8104 1.2386
rxLev+5FU 0.6020 1.6610 0.4766 0.7605
age 0.9974 1.0026 0.9897 1.0051
surg 1.2865 0.7773 1.0547 1.5693
obstruct 1.2381 0.8077 0.9884 1.5510
differentiationpoor 1.4271 0.7007 1.1267 1.8075
differentiationwell 1.0322 0.9688 0.7516 1.4176
node4 2.3158 0.4318 1.9144 2.8013
local_spreadmuscle 0.3244 3.0830 0.1966 0.5353
local_spreadserosa 0.5763 1.7353 0.3942 0.8424
local_spreadsubmucosa 0.3020 3.3111 0.1162 0.7849
Concordance= 0.667 (se = 0.012 )
Likelihood ratio test= 146.5 on 11 df, p=<2e-16
Wald test = 147.9 on 11 df, p=<2e-16
Score (logrank) test = 157.4 on 11 df, p=<2e-16anova(m4)Analysis of Deviance Table
Cox model: response is Surv(time, status)
Terms added sequentially (first to last)
loglik Chisq Df Pr(>|Chi|)
NULL -3040.3
rx -3028.1 24.3435 2 5.175e-06 ***
age -3026.8 2.5853 1 0.1078626
surg -3024.2 5.2257 1 0.0222552 *
obstruct -3022.7 3.0968 1 0.0784487 .
differentiation -3015.6 14.1005 2 0.0008672 ***
node4 -2978.2 74.7933 1 < 2.2e-16 ***
local_spread -2967.0 22.3477 3 5.522e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1cox_summary <- tidy(m4)
cox_summary# A tibble: 11 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 rxLev 0.00188 0.108 0.0174 9.86e- 1
2 rxLev+5FU -0.507 0.119 -4.26 2.07e- 5
3 age -0.00262 0.00392 -0.667 5.05e- 1
4 surg 0.252 0.101 2.49 1.29e- 2
5 obstruct 0.214 0.115 1.86 6.31e- 2
6 differentiationpoor 0.356 0.121 2.95 3.18e- 3
7 differentiationwell 0.0317 0.162 0.196 8.45e- 1
8 node4 0.840 0.0971 8.65 5.30e-18
9 local_spreadmuscle -1.13 0.256 -4.41 1.06e- 5
10 local_spreadserosa -0.551 0.194 -2.85 4.44e- 3
11 local_spreadsubmucosa -1.20 0.487 -2.46 1.40e- 2c_index_m4 <- concordance(m4)
cat("Concordance of the model with multivariate stepwise v_select:",c_index_m4$concordance)Concordance of the model with multivariate stepwise v_select: 0.6670357Test whether proportional hazard assumptions are met for model 4 predictors
cox.zph(m4) # final model with stepwise variable selection chisq df p
rx 0.0758 2 0.96283
age 0.0400 1 0.84140
surg 1.0410 1 0.30758
obstruct 4.7833 1 0.02874
differentiation 18.8075 2 8.2e-05
node4 10.0945 1 0.00149
local_spread 1.7889 3 0.61735
GLOBAL 37.2172 11 0.00011zph_test <- cox.zph(m4)
print(zph_test) chisq df p
rx 0.0758 2 0.96283
age 0.0400 1 0.84140
surg 1.0410 1 0.30758
obstruct 4.7833 1 0.02874
differentiation 18.8075 2 8.2e-05
node4 10.0945 1 0.00149
local_spread 1.7889 3 0.61735
GLOBAL 37.2172 11 0.00011# plot the Schoenfeld residuals
plot(zph_test)
Insight: Differentiation, node4 and obstruct variables did not meet proportional hazards assumption.
Stratify model by variables violating roportional hazard assumption
m5 <- coxph(Surv(time, status) ~ rx + age + surg + strata(obstruct) + strata(differentiation) + node4 +
local_spread, data = df)
summary(m5)Call:
coxph(formula = Surv(time, status) ~ rx + age + surg + strata(obstruct) +
strata(differentiation) + node4 + local_spread, data = df)
n= 929, number of events= 468
coef exp(coef) se(coef) z Pr(>|z|)
rxLev -0.002209 0.997793 0.108448 -0.020 0.98375
rxLev+5FU -0.494982 0.609582 0.119133 -4.155 3.25e-05 ***
age -0.001895 0.998107 0.003932 -0.482 0.62989
surg 0.263478 1.301449 0.101411 2.598 0.00937 **
node4 0.819875 2.270215 0.097645 8.396 < 2e-16 ***
local_spreadmuscle -1.160631 0.313288 0.256066 -4.533 5.83e-06 ***
local_spreadserosa -0.566254 0.567648 0.194214 -2.916 0.00355 **
local_spreadsubmucosa -1.268435 0.281272 0.487704 -2.601 0.00930 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
rxLev 0.9978 1.0022 0.8067 1.2341
rxLev+5FU 0.6096 1.6405 0.4826 0.7699
age 0.9981 1.0019 0.9904 1.0058
surg 1.3014 0.7684 1.0669 1.5876
node4 2.2702 0.4405 1.8748 2.7491
local_spreadmuscle 0.3133 3.1919 0.1897 0.5175
local_spreadserosa 0.5676 1.7617 0.3879 0.8306
local_spreadsubmucosa 0.2813 3.5553 0.1081 0.7316
Concordance= 0.672 (se = 0.014 )
Likelihood ratio test= 125.7 on 8 df, p=<2e-16
Wald test = 127.1 on 8 df, p=<2e-16
Score (logrank) test = 134 on 8 df, p=<2e-16cox.zph(m5) # final model with stratification by variables violating proportional hazard assumption chisq df p
rx 0.2291 2 0.8918
age 0.0102 1 0.9194
surg 1.4847 1 0.2230
node4 8.6990 1 0.0032
local_spread 2.0043 3 0.5715
GLOBAL 12.8524 8 0.1170zph_test <- cox.zph(m5)
print(zph_test) chisq df p
rx 0.2291 2 0.8918
age 0.0102 1 0.9194
surg 1.4847 1 0.2230
node4 8.6990 1 0.0032
local_spread 2.0043 3 0.5715
GLOBAL 12.8524 8 0.1170# plot the Schoenfeld residuals
plot(zph_test)
Insight: After model stratification by obstruct and differentiation, the proportional hazard assumption is met as the global p >0.05. Node4 slightly violates assumption, but the final model is not stratified by node 4 because the model concordance is attenuated when stratifying by node4.
Model comparision
library(knitr)
models <- list(m0, m1, m2, m3, m4, m5)
# Add descriptions for each model
descriptions <- c(
"Model 0 - Base model",
"Model 1 - Treatment",
"Model 2 - Full variables",
"Model 3 - Significant predictors in model 2",
"Model 4 - Selected stepwisely",
"Model 5 - model 4 Stratified"
)
# create a data frame to store results
results <- data.frame(Model = character(),
Description = character(),
AIC = numeric(),
BIC = numeric(),
C_Index = numeric(),
stringsAsFactors = FALSE)
# function to calculate and store metrics for each model
for (i in seq_along(models)) {
model <- models[[i]]
# Extract AIC and BIC
aic <- AIC(model)
bic <- BIC(model)
# add C-index
c_index <- concordance(model)$concordance
# append results to the data frame
results <- rbind(results, data.frame(
Model_variables = descriptions[i],
AIC = aic,
BIC = bic,
C_Index = round(c_index,3)
))
}
kable(results,caption = "Model evaluation matrics")| Model_variables | AIC | BIC | C_Index |
|---|---|---|---|
| Model 0 - Base model | 6080.552 | 6080.552 | 0.500 |
| Model 1 - Treatment | 6060.208 | 6068.505 | 0.554 |
| Model 2 - Full variables | 5958.758 | 6016.837 | 0.669 |
| Model 3 - Significant predictors in model 2 | 5954.501 | 5995.986 | 0.667 |
| Model 4 - Selected stepwisely | 5956.059 | 6001.692 | 0.667 |
| Model 5 - model 4 Stratified | 4757.394 | 4790.582 | 0.672 |
K-fold cross validation
library(survival)
library(boot) # for bootstrapping
library(survcomp) # to calculate c-index
library(caret)
set.seed(1234)
# Cox model
cox_model <- coxph(Surv(time, status) ~ rx + age + surg + strata(obstruct) + strata(differentiation) + node4 +
local_spread, data = df)
# calculate the original c-index
c_index_original <- concordance(cox_model)
cat("original c-index:", c_index_original$concordance, "\n")original c-index: 0.6718827 # create a function for calculating c-index in each fold using concordance()
cox_cindex <- function(train_data, test_data) {
fit <- coxph(Surv(time, status) ~ rx + age + surg + strata(obstruct) + strata(differentiation) + node4 + local_spread, data = train_data)
# Calculate concordance on test data
c_index <- concordance(fit, newdata = test_data)$concordance
return(c_index)
}
# perform 5-fold cross-validation with stratification
K <- 5
folds <- createFolds(c(df$status, df$differentiation, df$rx), k = K, list = TRUE, returnTrain = TRUE)
cv_c_indices <- sapply(folds, function(train_indices) {
train_data <- df[train_indices, ]
test_data <- df[-train_indices, ]
cox_cindex(train_data, test_data) # use the concordance function inside cox_cindex
})
# Print cross-validated c-indices
print(cv_c_indices) Fold1 Fold2 Fold3 Fold4 Fold5
0.7012768 0.6365716 0.6360538 0.6957962 0.6648189 # cross-validation c-indices
cat("cross-validated c-Indices for each fold:", cv_c_indices, "\n")cross-validated c-Indices for each fold: 0.7012768 0.6365716 0.6360538 0.6957962 0.6648189 cat("mean cross-validated c-Index:", mean(cv_c_indices), "\n")mean cross-validated c-Index: 0.6669035 # plot cross-validation c-indices
plot(cv_c_indices, type = "b", xlab = "Fold", ylab = "c-index", main = "c-index across folds")
Insight: The original model c-index (0.674) and mean cross-validation c-index (0.675) is very similar, suggesting the the final stratified model is stable and is not overfitting.
V. Results
Table 2. Univariate model: Survival after Chemotherapy for stage B/C Colon Cancer
| Treatment | Coefficient | Hazard ratio | 95% CI_upper | 95% CI_lower | P-value |
|---|---|---|---|---|---|
| Amisole (Lev) | -0.027 | 0.974 | 0.784 | 1.209 | 0.809 |
| Amisole + 5-FU | -0.372 | 0.690 | 0.546 | 0.870 | 0.002 |
Table 3. Multivariate model: Survival after Chemotherapy for stage B/C Colon Cancer
| Treatment | Coefficient | Hazard ratio | 95% CI_upper | 95% CI_lower | P-value |
|---|---|---|---|---|---|
| Amisole (Lev) | -0.011 | 0.989 | 0.795 | 1.231 | 0.923 |
| Amisole + 5-FU | -0.376 | 0.687 | 0.543 | 0.868 | 0.002 |
| Age | 0.007 | 1.007 | 0.999 | 1.015 | 0.069 |
| Surge | 0.244 | 1.276 | 1.042 | 1.562 | 0.018 |
| Obstruction of colon | 0.283 | 1.327 | 1.057 | 1.667 | 0.015 |
| Differentiat ion_poor | 0.374 | 1.453 | 1.145 | 1.844 | 0.002 |
| Differentiat ion_well | 0.069 | 1.072 | 0.774 | 1.483 | 0.677 |
| More than 4 nodes (+) | 0.930 | 2.534 | 2.089 | 3.074 | 3.75 x 10-21 |
| Local spread_muscle | -0.996 | 0.370 | 0.225 | 0.606 | 7.85 x 10-5 |
| Local spread_serosa | -0.501 | 0.606 | 0.414 | 0.886 | 0.010 |
| Local spread_submucosa | -1.322 | 0.267 | 0.093 | 0.763 | 0.014 |
References
[1]
Terry M. Therneau and Patricia M. Grambsch, Modeling survival data: Extending the Cox model. New York: Springer, 2000.
[2]
T. M. Therneau, A package for survival analysis in r. 2024. Available: https://CRAN.R-project.org/package=survival
[3]
R Core Team, R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, 2023. Available: https://www.R-project.org/
[4]
A. Kassambara, M. Kosinski, and P. Biecek, Survminer: Drawing survival curves using ’ggplot2’. 2021. Available: https://CRAN.R-project.org/package=survminer
[5]
W. N. Venables and B. D. Ripley, Modern applied statistics with s, Fourth. New York: Springer, 2002. Available: https://www.stats.ox.ac.uk/pub/MASS4/
[6]
A. C. Davison and D. V. Hinkley, Bootstrap methods and their applications. Cambridge: Cambridge University Press, 1997. Available: doi:10.1017/CBO9780511802843
[7]
Angelo Canty and B. D. Ripley, Boot: Bootstrap r (s-plus) functions. 2024.